Struggling to install, dual boot with Windows 10

My PC came with Windows 8, so I assume it has the UEFI thingy?
I previously had Ubuntu installed but some time ago I reset Windows and deleted the Ubuntu partitions.
I've followed the tutorial on zorinos.com but when it finishes installing and goes to restart, it boots through the USB stick I'm using and I get the installation options again. There isn't a grub or anything like there was for Ubuntu. I've tried boot repair but if I launch it in non-UEFI mode it says it must be launched in UEFI. However if I attempt to launch it through UEFI I get that black screen with white text saying "GNU Grub version 2.02 minimal bash-like line editing is supported", which is also what happens if I use the boot option that says ubuntu and a bunch of numbers or something after doing the zorin installation.
P.S.: For some reason when installing zorin it says that I have Windows 8, not Windows 10.
I have tried the automatic install and manual install with the /, /swap and /home stuff (I had done it this way for Ubuntu I think)

I'd appreciate some help, thanks.

OK, Was this notebook with 8 and you upgraded to 10? I get the same GRUB issue as you and that might be down to the need to format partitions to GPT mode for UEFI. I use my BIOS to boot to the required drive, otherwise I would have to remove my Windows 7 64-bit from the hotbay. Have you used Windows Disk Management to make space for Zorin first by shrinking C:\? Also do you have factory reset partition/s still present? GPT partitions allow more than 4 primary but if Notebook not using GPT then you are restricted to 4 Primary Partitions, regardless of how many drives you have.

Yes, I had shrinked C: by 125GB to have space for Zorin. I don't understand much about this, so here's a screenshot of my partition manager.
(The unallocated space becomes allocated after the installation)
And yes, it originally came with Windows 8, I upgraded to Windows 10 right when Microsoft released the free update.
EDIT: It's an ASUS ROG GL550JK, by the way

Sigh! It is a pity your column divider has not been moved to the right of the 'Status' column but I am pretty sure that those we cannot see are Primary Partitions also - that is why you can't boot - you are going to have to make a painful decision of deleting one or the other. Is there any way to create backup media of 'restore' and 'recovery'? Your only other option would be to start again, revert back to the drive's original layout and backup the entire system, then look at deleting either 'restore' or 'recovery'.